# Pair of Opposite Sides of a Parallelogram are Equal and Parallel

Here we will discuss about one of the important geometrical property of parallelogram.

A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel

Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR.

To prove: PQRS is a parallelogram.

Construction: Join PR and QS such that they intersect at O.

Proof:

 Statement Reason In ∆OPQ and ∆ORS,1. ∠OPQ = ∠ORS 1. PQ ∥ SR and PR is a transversal. 2. ∠POQ = ∠ROS 2. Opposite angles are equal. 3. PQ = RS 3. Given. 4. ∆OPQ ≅ ∆ORSTherefore, OP = OR, OQ = OS.In ∆OPS and ∆OQR, 4. By AAS criterion of congruency.CPCTC 5. OP = OC, OQ = OS, ∠POS = ∠QOR 5. By statement 4 and reason 2. 6. ∆OPS ≅ ∆OQRTherefore, PS = QR, ∠OPS= ∠ORQ 6. By SAS criterion of congruency.CPCTC 7. PS ∥QR. 7. Alternate angles are equal. 8. PQRS is a parallelogram (Proved). 8. PQ ∥ SR and statement 7.

Corollary: In a parallelogram, each pair of opposite sides are parallel as well as equal.

From Pair of Opposite Sides of a Parallelogram are Equal and Parallel to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

Jul 12, 24 03:08 PM

The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerato…

2. ### Worksheet on Fractions | Questions on Fractions | Representation | Ans

Jul 12, 24 02:11 PM

In worksheet on fractions, all grade students can practice the questions on fractions on a whole number and also on representation of a fraction. This exercise sheet on fractions can be practiced

3. ### Fraction in Lowest Terms |Reducing Fractions|Fraction in Simplest Form

Jul 12, 24 03:21 AM

There are two methods to reduce a given fraction to its simplest form, viz., H.C.F. Method and Prime Factorization Method. If numerator and denominator of a fraction have no common factor other than 1…

4. ### Conversion of Improper Fractions into Mixed Fractions |Solved Examples

Jul 12, 24 12:59 AM

To convert an improper fraction into a mixed number, divide the numerator of the given improper fraction by its denominator. The quotient will represent the whole number and the remainder so obtained…